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Brown(Leonardo) 1/13/05 3:05 PM Page 16

“Many of history’s famous word problems first appeared in my book: mixtures, percentage, discount, the wolf-goat-cabbage, the hen-and-a-half laying an egg-and-a-half in a day-and-a-half, etc.” A Conversation with Leonardo Pisano Ezra Brown Virginia Tech

hree in the morning, too much coffee, and hours of por- let’s just say that I was born in the late 1170s. ing over the new English translation of ’s 1202 MH: What was it like growing up in ? Tmasterwork Liber Abaci—that could be the only expla- nation for why the great man himself materialized in the only LP: Pisa, , , and were the Republiche vacant chair in the room. After I recovered my composure, we Marinare, the four . It was a trade alliance; had this conversation: they had divided up the trade routes to the Mediterranean countries and beyond. So, Pisa was a pretty lively place, with Math Horizons: Are you the illustrious Fibonacci? the pepper, leather, and fur trades and so many people coming Leonardo Pisano: You talkin’ to me? and going. . .a great place to grow up in. Rough, at times, and filthy-dirty by your standards, but a great place. Pisa never was MH: Well, yes. . .at least, I thought you were . . . very big, by the way; its population in my time was maybe LP: . . .the thirteenth century mathematician Leonardo Pisano? 10,000. We had no king or prince who ruled over us; we were Yes, I am. . .or rather, I was. . .time gets muddled after a few a republic. You know what a republic is, don’t you, Pal? centuries, you know. MH: Certainly. MH: But you are now universally known as Fibonacci, aren’t LP: Well, then, because times were good, the merchants pret- you? ty much ruled the roost. They put kids to work early in those LP: Look, Pal, that tag was laid on me by this nineteenth cen- days; labor was scarce, and children were needed in the shops. tury historian Guillaume Libri. Nobody called me Fibonacci in My family figured out that I wasn’t just an ordinary kid, but my lifetime. In my time, not everyone had last names. My one with a head on his shoulders, so I was taught to make name meant “Leonardo of Pisa,” my father was Guilielmo and change and read a ledger. My family was big on literacy. “Bonaccio” is a family name. But hey, forgive and forget. At You’ll never guess who taught me to read. least, they haven’t forgotten me. MH: Who was it? MH: Yes, well, Leonardo, mathematicians and students of our LP: My grandmother. She had been taught to read by the son time are really curious about your life and your work, so may of a family friend. The son had been to the University of I ask you some questions? Bologna, was passing through Pisa, and stayed with Great LP: Go right ahead, Pal. What do you want to know? Grandpa for a while. He was quite taken with Grandma, who MH: First of all, when were you born? was a small child, and would patiently answer her questions about the world. He had a book of poems, and would read to LP: Born? Who knows. It was considered bad luck to write the her, and eventually, she figured out the trick of reading and name of a child until you reached a certain age. Record keep- began to follow along. Fifteen years later, the student came ing was spotty even in the best of times, and Pal, those were back and married her. He’s my grandpa. Anyway, she taught the best of times for Pisa. They began building the bell me to read out of that same book of poetry. in the 1170s, but stopped after putting up only three stories when they realized what rotten ground they’d chosen for the MH: Tell us about your education. site. Anyway, I always remember at least part of the tower LP: We began religious training as six-year-olds, but as I said, being there. According to the preface of Liber Abaci, Pop sent even before that I learned to read. Hanging around merchants, word for me to come to the North African seacoast port of peddlers, money-lenders, etc. I learned your basic “street Bugia, sometime before I was eighteen. That was in 1192, so math.” For instance, if you were working in a shop and an

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important buyer came in from , he might not have what I wrote. As long as I was introducing Indian figures to the Pisan money. You had to know rates of exchange and convert merchants of Pisa and the Maritime Republics, it would have between many different currencies. I put quite a few problems been even better to come up with some powerful new notation. involving foreign exchange in Liber Abaci, you know. Mer- But eight hundred years of hindsight, and all that—anyway, I chants had to work quickly and accurately, juggling money wrote it, and revised it in 1228. People now say that most of it from Amalfi, Byzantium, , Venice, etc. In Bugia, I was wasn’t original, and they’re right, but so what? That wasn’t introduced to the wonders of Indian figures, including the why I wrote it. It was a big hit in my time. Of course, the Firen- zephir, and eventually the al-jabr. My father hired a skilled zi, the Florentine merchants objected to using the Indian fig- teacher to instruct the Pisan merchants in the Indian figures ures, claiming that they invited fraud and forgery. What people and I was included in that instruction. I think it was Pop’s plan forget, Pal, is that and Pisa were never on the best of for me to be a merchant, too. But the haggling and worrying terms, and for them to use a method suggested by a about making sales and fretting about prices of goods and deal- Pisano…well, I wasn’t surprised that they passed laws against ing with stubborn customers—that wasn’t for me. So I became the use of Indian figures, laws that lasted until the fifteenth a teacher and a writer. century, when printing with movable type came along. MH: How did that happen, and how did you come to write the MH: You wrote a Book of Squares, so why not a Book of Liber Abaci? Cubes? LP: The reasons are right there in the prologue, but I guess LP: Why didn’t I write a Liber Cuborum? Hey, Pal, that’s an what you really want to know is, why me? I was the right man easy one. I’ve got a reputation to maintain, and I didn’t con- for the job—that’s why. You see, I learned quickly and could sider myself an expert...besides, it was gonna be hard enough explain the use of Indian figures in a way that merchants and for people to read Liber Quadratorum, even though squares other people could understand. There was clearly a need for are really not that tough. One more thing: in those days, I good instruction, and when I returned to Pisa after traveling the didn’t believe that you could solve cubic equations using al- Mediterranean world, it seemed like a good idea to write a jabr...300 years later, a bunch of Giovannis-come-lately proved book on the Indian figures and the al-jabr, complete with many me wrong—and they were all from my part of the world. examples that merchants would understand. Many of history’s MH: Did you leave any unsolved problems that you’d like to famous word problems are in there in one form or another: return to in the modern day? mixtures, percentage, discount, the wolf-goat-cabbage, the LP: chicken-and-a-half laying an egg-and-a-half in a day-and-a- Sure, but most of them got solved in the meantime. The half, etc. The man-who-traveled-in-twelve-cities problem is an sixteenth century Italian lads got the cubic (and quartic) for- example of exponential decay, 450 years before Newton’s Law mulas, Newton extended and generalized my approximations, of Cooling. I included problems with large numbers, too, so and Fermat and Euler went through my number theory like that people could see the power of Indian figures. By the way, Alexander the Great. In your time, a fellow named Tunnell solved the congruous number problem, recursion has been the largest number in the book is 364. I also included proofs in taken over by the wizards with those magic machines, and iter- the Euclidean manner, so that learned scholars would believe ation has literally turned to Chaos! What’s left for a kid from Pisa? Precious little. MH: What do you think about the vast amount of research that has been done on your Fibonacci numbers and their general- izations? LP: I like it! It’s very gratifying to see your name everywhere, Pal; there’s even an entire society devoted to the Rabbit Num- bers. But the most startling wrinkle on the original problem happened in the last five years. It’s Divakar Viswanath’s amaz-

ing work on the sequence defined by t1 = t2 = 1 and for n > 2, tn = ± tn–1 ± tn–2, where the signs are independent and either + or – with probability 0.5. He proved that, with probability 1, n limnn→∞ t = 1.13198824… Some people are calling this number a new constant that rivals the Golden Ratio ϕ =+()152/,the limit of the ratio of consecutive Fibonacci numbers. Hey, I’m impressed!

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LP: Not one single area, Pal. There wasn’t anything there to build on. The Crusades brought Europe in contact with the much more advanced cultures of the East, and scholars had begun some impressive translation projects, but the only rea- son I knew about Euclid was that there were Arabic transla- tions. Here’s what the eight centuries of European mathemat- ics after Pappus and before me produced: zilch. MH: Which of your many achievements do you consider your best work? LP: You don’t ask easy questions, Pal. I like the proof in Liber Quadratorum of the formula for 12+22+32++ n2 that uses what you now call telescoping sums. As I said before, the Rab- bit Problem became a whole lot bigger than just a simple counting problem, because recursion and iteration are hot top- ics in your time. In Chapter 2 of Liber Abaci, I mentioned Casting Out Thirteens and other checks on calculations; before my time, it was only Casting Out Nines. This was modular Cartoon by John Johnson arithmetic 600 years before Gauss’ Disquisitiones Arithmeti- cae; this was a detailed error-detecting scheme 750 years before Hamming’s parity checks and error-correcting codes. I MH: Did you consider yourself a “pure” or an “applied” math- like what I did with the congruous numbers problem, finding a ematician, or did you never even consider such a distinction? rational number x such that x2 +5 and x2–5 are both squares, LP: Let’s put it this way. I wrote Liber Abaci as a practical text which Euler extended way beyond what I did. The first for merchants and as a way to spread the word about the Indi- approximation to the cube root of a noncube had been around an figures and al-jabr. I wrote Liber Quadratorum for my own quite a while—the Chinese had it, I think—but I was the first benefit. I wrote up Flos as a record of the Imperial contest, so to spell out the second approximation. It’s the next step in nobody’d forget. You figure it out, Pal. Newton’s method, but in my day, who knew about Newton? MH: Does it surprise you that modern mathematicians con- Then there’s that six-sexagesimal-place approximation to the sider algebra and numerical analysis to be largely separate solution of x3 + 2x2 + 10x = 20 that I showed to Frederick II. . subjects? .the looks of amazement on his face and on Giovanni di Paler- LP: Not really. has grown so large that you have mo’s ugly mug are still with me after over 750 years! to split things up. Nobody since Hilbert and Poincaré has come MH: Finally, do you have any words of wisdom for our close to understanding it all, or even most of it. students? MH: Did you anticipate that the solution of the quintic would LP: That’s easy: join the MAA and subscribe to Math Hori- present so much difficulty? zons! Listen, Pal, I’m late for a chat with , LP: Actually, I never thought anyone would solve the cubic, Leonhard Euler, and Leonard Carlitz. Around those three, I’m let alone the quintic. always in the dumbest quarter of the room, so arrivederci, Pal! MH: Why would a king be interested in meeting a mathemati- Then he disappeared. The next morning, my wife asked me, “I cian? heard voices downstairs last night; who were you talking with?” Well, what would you say? LP: The Holy Roman Emperor Frederick II was a complex man. He could be ruthless and cruel, but he was also interest- Suggested Reading ed in art, music, and the intellect—a patron of the arts, you Start with the Master and read L. E. Sigler’s marvelous might say. When he traveled to Pisa, his advisor Giovanni di English translations, Fibonacci’s Liber Abaci, Springer-Ver- mentioned that Leonardo Pisano was someone worth lag, New York, 2002, and “The Book of Squares (English meeting, so they arranged for me to be presented at Freder- Translation of Liber Quadratorum),” Academic Press, New ick’s court. York, 1987. The references in these books are rich in informa- MH: In your time, were there any areas of mathematics in tion about Leonardo and his work. The website which Europe excelled, but Indian, Arabic, or other Eastern http://www.maa.org/devlin/devlin_3_ mathematicians lacked? 99.html is a readable account of Divakar Viswanath’s work.

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